Compact phase-generator mixture

Let $K$ be a nonempty compact subset of ${\mathbb{R}}^d$, $d\ge 1$, with its Borel sigma-algebra, and let $a:K\to \mathbb{R}$ be continuous. The convex hull $\mathrm{conv}K$ is the set of finite convex combinations of points of $K$. A phase generator is a point $y\in K$ with declared energy $a(y)$. A mixture is a Borel probability measure $P$ on $K$. Its barycenter and mean energy are the coordinatewise Lebesgue integrals $\bar{y}(P):={\int }_{K}ydP(y)\in {\mathbb{R}}^d,\bar{a}(P):={\int }_{K}a(y)dP(y)\in \mathbb{R}\mathrm{.}$ These integrals are finite because each coordinate and $a$ are bounded on compact $K$. For prescribed mean $x\in \mathrm{conv}K$, the mixture energy is $a_{\mathrm{m}\mathrm{i}\mathrm{x}}(x):=\inf \{\bar{a}(P):P\text{ is a Borel probability on }K,\bar{y}(P)=x\}\mathrm{.}$ The admissible set is nonempty since every finite convex combination of points of $K$ is the barycenter of a finite atomic probability. The definition does not assert that the infimum is attained; the theorem on this page gives a compact continuous sufficient condition.

The definition concerns a mathematical probability mixture and an assigned scalar energy. It does not assert that physical phases can be prepared, exchanged, or brought to equilibrium.